# Math 380 - Probability Theory Spring 2017

Instructor: Elizabeth Meckes

Office: Yost 208

Phone: 368-5015

Email: ese3 [at] cwru.edu

Office Hours: MWF, 11:30 -- 12:30

Textbook: We'll be using a draft of the soon-to-be-published Introduction to Probability by David Anderson, Timo Seppäläinen, and Benedik Valkó. It is posted in the Canvas site for this course.

Course web page:
http://www.cwru.edu/artsci/math/esmeckes/math380/.

All course information (including the homework assignments!) is posted here; Canvas is used only for grades and a place to post the text book.

Topics and rough schedule:
The schedule will be roughly as follows:

TopicsBook chaptersWeeks
Random outcomes and the rules of probability
Conditional probability and independence
1,2 1-3
Random variables 3 4
Approximations of the binomial distribution 4 5-6
Transformations of random variables
Joint distributions
5,6 7-8
Sums and symmetry 7 9
Expectation and variance 8 10-11
Limit theorems 9 12-13
Conditional distributions 10 14

Attendance:
You're supposed to come. (To every class.)

Reading the book and attending (and actively learning from!) the lectures are complementary, and it's important to do both. Before each class, please read the section to be covered in the next lecture (we'll go through the book in order — I'll announce any exceptions in class). You will be placed in a group of four at the beginning of the semester; each class will start with a short group quiz based on the material you read in preparation for class.

Homework Problems:
How much you work on the homework problems is probably the single biggest factor in determining how much you get out of the course. If you are having trouble with the problems, please come ask for help; you will learn much more (and probably get a rather better grade) if you figure out all of the homework problems, possibly with help in office hours or from your classmates, than if you do them alone when you can and skip the ones you can't. Students are welcome to work together on figuring out the homework, but you should write up the solutions on your own.

Each lecture has specific homework problems associated to it, as listed in the chart below. I strongly suggest doing the homework the same day as the corresponding lecture (see in particular this figure titled "The value of rehearsal after a lecture"). Homework will be collected weekly.

The homework is meant to be a mix of relatively straightforward exercises and harder problems. Don't worry too much if you find some of it hard, but do continue to struggle with it; that's the way you learn.

The next stage after the struggle of figuring out a problem is writing down a solution; you learn a lot here, too. The homework assignments are writing assignments, and what you turn in should be polished (edited!) English prose with well-reasoned, complete arguments. I should be able to give your solutions to another student who has never thought about the problems (or did, but didn't figure them out), and she should be able to read and understand them.

Individual quizzes:
There will be four hour-long quizzes throughout the term. These are closed book, closed notes. The tentative dates are: February 6, February 27, April 3, May 1.

• Group quizzes 5%
• Homework 15%
• Midterm exams 55%
• Final exam 25%

Forget What You Know About Good Study Habits appeared in the Times in Fall 2010. It offers some advice about studying based on current pedagogical research.

Teaching and Human Memory, Part 2 from The Chronicle of Higher Education in December 2011. Its intended audience is professors, but I think it's worth it for students to take a look as well.

Assignments:
Howework is posted below.

W 1/1811.1, 1.2, 1.211/25Sections 1.1, 1.2
F 1/20quiz11.5, 1.7, 1.9, 1.10, 1.221/25Sections 1.3, 1.4
M 1/23quiz11.12, 1.14, 1.15, 1.251/25Section 1.5
W 1/25quiz11.17, 1.18, 1.19, 1.32, 1.502/1Section 2.1
F 1/27quiz22.5, 2.6, 2.7, 2.8, 2.282/1Sections 2.2, 2.3
M 1/30quiz22.9, 2.10, 2.11, 2.342/1Section 2.4
W 2/1quiz22.13, 2.14, 2.15, 2.17, 2.372/8none
F 2/3quiz22.19, 2.20, 2.22, 2.392/82.5
M 2/6Exam 12.5
W 2/8quiz22.24, 2.26, 2.59, 2.642/153.1
F 2/10quiz33.2, 3.3, 3.4, 3.262/153.2
M 2/13quiz33.6, 3.7, 3.8, 3.202/153.3
W 2/15quiz33.11, 3.12, 3.13, 3.302/223.4, 3.5
F 2/17quiz33.23, 3.27, 3.35, 3.602/224.1
M 2/20quiz3/43.18, 3.37, 3.39, 4.13, 4.152/224.2
W 2/22quiz44.3, 4.17, 4.203/14.3
F 2/24quiz44.4, 4.5, 4.253/1none
M 2/27Exam 24.4
W 3/1quiz44.8, 4.28, 4.30, 4.343/84.5
F 3/3quiz44.11, 4.36, 4.45, 4.483/84.6, 5.1 (through pg. 165)
M 3/6quiz44.37, 4.40, 4.46, 4.493/85.1
W 3/8quiz55.2, 5.4, 5.5, 5.7, 5.143/225.2
F 3/10quiz55.8, 5.9, 5.15, 5.21, 5.293/226.1
M 3/20quiz66.2, 6.4, 6.18, 6.203/226.2
W 3/22quiz66.5, 6.6, 6.7, 6.22, 6.353/296.3
F 3/24quiz66.8, 6.11, 6.13, 6.25, 6.283/296.4
M 3/27quiz66.15, 6.16, 6.45, 6.493/297.1
W 3/29quiz77.1, 7.3, 7.4, 7.11, 7.124/5none
F 3/31quiznonenone
M 4/3Exam 37.2
W 4/5quiz77.6, 7.7, 7.8, 7.17, 7.284/128.1
F 4/7quiz88.1, 8.3, 8.5, 8.15, 8.174/128.2
M 4/10quiz88.6, 8.18, 8.27, 8.304/128.4
W 4/12quiz88.9, 8.10, 8.28, 8.344/198.4
F 4/14quiz88.12 (hint: check covariances), 8.42, 8.43, 8.454/199.1
M 4/17quiz99.1, 9.2, 9.3, 9.74/199.2, 9.3
W 4/19quiz99.4, 9.5, 9.8, 9.10, 9.134/26 9.3, 9.4
F 4/21quiz99.14, 9.16, 9.17, 9.194/26 10.1
M 4/24quiz1010.2, 10.3, 10.12, 10.154/26 10.3
W 4/26quiz1010.14, 10.16, 10.17, 10.24uncollected (but good midterm prep!) none

Final exam information

The final exam will be Wednesday, May 10 at 8 am (sorry!) in the usual classroom. It will cover all the course material.

Definitions

You will be asked for complete, precise definitions of about four of the following terms, including context (e.g., what kind of thing can have a Poisson distribution?). Definitions should be complete English sentences.

Remember that a complete mathematical definition leaves no room for ambiguity: if you give me a definition of what it means for a random variable to be discrete, I need to be able to use it to decide (correctly!) whether or not any random variable I ever meet is discrete.
• sample space
• event
• probability measure
• probability space
• sampling with replacement
• sampling without replacement
• inclusion/exclusion
• random variable
• distribution of a random variable
• probability mass function
• density
• conditional probability of A given B
• independent events
• independent random variables
• Bernoulli random variable
• binomial distribution
• geometric distribution
• uniform distribution
• Gaussian distribution
• cumulative distribution function
• expected value
• median
• variance
• equality in distribution
• weak law of large numbers
• strong law of large numbers
• Poisson process
• moments
• moment generating function
• equality in distribution
• joint distribution
• joint probability mass function
• joint density
• independent random variables
• jointly continuous random variables
• joint cumulative distribution function
• convolution (discrete case)
• convolution (continuous case)
• Poisson distribution
• Exponential distribution
• exhangeable
• i.i.d.
• sample mean
• sample variance
• covariance
• correlation
• uncorrelated
• positively correlated
• negatively correlated
• Markov's inequality
• Chebychev's inequality
• conditional mass function
• conditional expectation (given an event)
• conditional expectation (given the value of a random variable)
• conditional expectation (given a random variable)

Midterm 4 information

The next midterm will be Monday, May 1 in class. It will cover all the course material covered between April 5 and April 26.

Definitions

You will be asked for complete, precise definitions of about four of the following terms, including context (e.g., what kind of thing can have a Poisson distribution?). Definitions should be complete English sentences.

Remember that a complete mathematical definition leaves no room for ambiguity: if you give me a definition of what it means for a random variable to be discrete, I need to be able to use it to decide (correctly!) whether or not any random variable I ever meet is discrete.
• Poisson distribution
• Exponential distribution
• exhangeable
• i.i.d.
• sample mean
• sample variance
• covariance
• correlation
• uncorrelated
• positively correlated
• negatively correlated
• Markov's inequality
• Chebychev's inequality
• conditional mass function
• conditional expectation (given an event)
• conditional expectation (given the value of a random variable)
• conditional expectation (given a random variable)

Midterm 3 information

The next midterm will be Monday, April 3 in class. It will cover all the course material covered between March 1 and March 31.

Definitions

You will be asked for complete, precise definitions of about four of the following terms, including context (e.g., what kind of thing can have a Poisson distribution?). Definitions should be complete English sentences.

Remember that a complete mathematical definition leaves no room for ambiguity: if you give me a definition of what it means for a random variable to be discrete, I need to be able to use it to decide (correctly!) whether or not any random variable I ever meet is discrete.
• Poisson distribution
• Exponential distribution
• Poisson process
• moments
• moment generating function
• equality in distribution
• joint distribution
• joint probability mass function
• joint density
• independent random variables
• jointly continuous random variables
• joint cumulative distribution function
• convolution (discrete case)
• convolution (continuous case)